Showing papers on "Pressure gradient published in 1998"

Magnetohydrodynamic stability of tokamak edge plasmas

Abstract: A new formalism for analyzing the magnetohydrodynamic stability of a limiter tokamak edge plasma is developed. Two radially localized, high toroidal mode number n instabilities are studied in detail: a peeling mode and an edge ballooning mode. The peeling mode, driven by edge current density and stabilized by edge pressure gradient, has features which are consistent with several properties of tokamak behavior in the high confinement “H”-mode of operation, and edge localized modes (or ELMs) in particular. The edge ballooning mode, driven by the pressure gradient, is identified; this penetrates ∼n1/3 rational surfaces into the plasma (rather than ∼n1/2, expected from conventional ballooning mode theory). Furthermore, there exists a coupling between these two modes and this coupling provides a picture of the ELM cycle.

Direct numerical simulation of a separated turbulent boundary layer

Abstract: A separated turbulent boundary layer over a flat plate was investigated by direct numerical simulation of the incompressible Navier–Stokes equations. A suction-blowing velocity distribution was prescribed along the upper boundary of the computational domain to create an adverse-to-favourable pressure gradient that produces a closed separation bubble. The Reynolds number based on inlet free-stream velocity and momentum thickness is 300. Neither instantaneous detachment nor reattachment points are fixed in space but fluctuate significantly. The mean detachment and reattachment locations determined by three different definitions, i.e. (i) location of 50% forward flow fraction, (ii) mean dividing streamline (ψ=0), (iii) location of zero wall-shear stress (τw=0), are in good agreement. Instantaneous vorticity contours show that the turbulent structures emanating upstream of separation move upwards into the shear layer in the detachment region and then turn around the bubble. The locations of the maximum turbulence intensities as well as Reynolds shear stress occur in the middle of the shear layer. In the detached flow region, Reynolds shear stresses and their gradients are large away from the wall and thus the largest pressure fluctuations are in the middle of the shear layer. Iso-surfaces of negative pressure fluctuations which correspond to the core region of the vortices show that large-scale structures grow in the shear layer and agglomerate. They then impinge on the wall and subsequently convect downstream. The characteristic Strouhal number St=fδ*in/U0 associated with this motion ranges from 0.0025 to 0.01. The kinetic energy budget in the detachment region is very similar to that of a plane mixing layer.

A Wall-Distance-Free k-ε Model With Enhanced Near-Wall Treatment

Abstract: We evaluate a wall-distance-free low-Re κ-e turbulence closure model which Incorporates an extra source term in the e transport equation designed to increase the level of e in nonequilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, including those involving separated flows regions. Two such cases are used here to test the model: one in the low speed flow regime, the other a supersonic one. Comparisons with experimental data and with an earlier version of the κ-e model, as well as with a variant of the κ-ω model (both also wall-distance-free) reveal that the proposed model enables improved prediction of adverse pressure gradient flows relative to more traditional κ-e models

Investigation of controls on secondary circulation in a simple confluence geometry using a three-dimensional numerical model

Abstract: Recent research into river channel confluences has identified confluence geometry, and particularly bed discordance, as a control on confluence flow structures and mixing processes, and this has been illustrated using both field measurements in natural confluences and laboratory measurements of simplified confluences. Generalization of the results obtained from these experiments is limited by the number of confluence geometries that can be examined in a reasonable amount of time. This limitation may be overcome by numerical models, in which confluence geometry is more readily varied, and data acquired more rapidly. This paper aims to: (i) validate the application of a three-dimensional numerical model to a simple confluence geometry; (ii) simulate the effects of different boundary condition values upon flow structures; and (iii) interpret the implications of these simulations for river channel confluence dynamics. The model used in this research solves the three-dimensional form of the Navier–Stokes equations and is used to simulate the flow in a parallel confluence of unequal depth channels and to investigate the effect of different combinations of velocity and depth ratio between the two tributaries. The results generally agree with empirical evidence that secondary circulation is generated in the absence of streamline curvature, but only for specific combinations of depth and velocity ratio. This research shows how understanding of the interaction of these controls is enhanced if pressure gradients are considered. The velocity ratio is the prime determinant of the cross-stream pressure gradient that initiates cross-stream velocities. However, for significant secondary circulation to form, cross-stream velocities must lead to significant transfer of fluid in the cross-stream direction. This depends on the vertical extent of the cross-stream pressure gradient which is controlled by the depth ratio. In this study, strong secondary circulation occurred for a depth differential of 25% or more, as long as the velocity in the shallower tributary was at least as great as that in the deeper channel. This provides an important context for interpretation of previous work and for the design of new experiments in both the field and the laboratory. © 1998 John Wiley & Sons, Ltd.

Scaling studies of the high mode pedestal

Abstract: The structure and scaling of the H-mode pedestal are examined for discharges in the DIII-D tokamak. For typical conditions, the pedestal values of the ion and electron temperatures T{sub i} and T{sub e} are comparable. Measurements of main ion and C{sup 6+} profiles indicate that the ion pressure gradient in the barrier is 50%--100% of the electron pressure gradient for deuterium plasmas. The magnitude of the pressure gradient in the barrier often exceeds the predictions of infinite-n ballooning mode theory by a factor of two. Moreover, via the bootstrap current, the finite pressure gradient acts to entirely remove ballooning stability limits for typical discharges. For a large dataset, the width of the pressure barrier {delta} is best described by the dimensionless scaling {delta}/R {proportional_to} ({beta}{sub pol}{sup ped}){sup 0.4} where ({beta}{sub pol}{sup ped}) is the pedestal value of poloidal beta and R is the major radius. Scalings based on the poloidal ion gyroradius or the edge density gradient do not adequately describe overall trends in the data set and the propagation of the pressure barrier observed between edge-localized modes. The width of the T{sub i} barrier is quite variable and is not a good measure of the width of the pressure barrier.

Direct simulation of transition in an oscillatory boundary layer

Abstract: Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

The structure of wall-pressure fluctuations in turbulent boundary layers with adverse pressure gradient and separation

Abstract: Space–time correlations and frequency spectra of wall-pressure fluctuations, obtained from direct numerical simulation, are examined to reveal the effects of pressure gradient and separation on the characteristics of wall-pressure fluctuations. In the attached boundary layer subjected to adverse pressure gradient, contours of constant two-point spatial correlation of wall-pressure fluctuations are more elongated in the spanwise direction. Convection velocities of wall-pressure fluctuations as a function of spatial and temporal separations are reduced by the adverse pressure gradient. In the separated turbulent boundary layer, wall-pressure fluctuations are reduced inside the separation bubble, and enhanced downstream of the reattachment region where maximum Reynolds stresses occur. Inside the separation bubble, the frequency spectra of wall-pressure fluctuations normalized by the local maximum Reynolds shear stress correlate well compared to those normalized by free-stream dynamic pressure, indicating that local Reynolds shear stress has more direct influence on the wall-pressure spectra. Contour plots of two-point correlation of wall-pressure fluctuations are highly elongated in the spanwise direction inside the separation bubble, implying the presence of large two-dimensional roller-type structures. The convection velocity determined from the space–time correlation of wall-pressure fluctuations is as low as 0.33U0 (U0 is the maximum inlet velocity) in the separated zone, and increases downstream of reattachment.

The effects of a favourable pressure gradient and of the Reynolds number on an incompressible axisymmetric turbulent boundary layer. Part 1. The turbulent boundary layer

Abstract: The effects of a favourable pressure gradient (K[les ]4×10−6) and of the Reynolds number (862[les ]Reδ2[les ]5800) on the mean and fluctuating quantities of four turbulent boundary layers were studied experimentally and are presented in this paper and a companion paper (Part 2). The measurements consist of extensive hot-wire and skin-friction data. The former comprise mean and fluctuating velocities, their correlations and spectra, the latter wall-shear stress measurements obtained by four different techniques which allow testing of calibrations in both laminar-like and turbulent flows for the first time. The measurements provide complete data sets, obtained in an axisymmetric test section, which can serve as test cases as specified by the 1981 Stanford conference.Two different types of accelerated boundary layers were investigated and are described: in this paper (Part 1) the fully turbulent, accelerated boundary layer (sometimes denoted laminarescent) with approximately local equilibrium between the production and dissipation of the turbulent energy and with relaxation to a zero pressure gradient flow (cases 1 and 3); and in Part 2 the strongly accelerated boundary layer with ‘inactive’ turbulence, laminar-like mean flow behaviour (relaminarized), and reversion to the turbulent state (cases 2 and 4). In all four cases the standard logarithmic law fails but there is no single parametric criterion which denotes the beginning or the end of this breakdown. However, it can be demonstrated that the departure of the mean-velocity profile is accompanied by characteristic changes of turbulent quantities, such as the maxima of the Reynolds stresses or the fluctuating value of the skin friction.The boundary layers described here are maintained in the laminarescent state just up to the beginning of relaminarization and then relaxed to the turbulent state in a zero pressure gradient. The relaxation of the turbulence structure occurs much faster than in an adverse pressure gradient. In the accelerating boundary layer absolute values of the Reynolds stresses remain more or less constant in the outer region of the boundary layer in accordance with the results of Blackwelder & Kovasznay (1972), and rise both in the vincinity of the wall in conjunction with the rising wall shear stress and in the centre region of the boundary layer with the increase of production.

Mobile layer in oscillatory sheet flow

Abstract: Measurements are reported of mobile layer thickness and the velocity and concentration distributions within the mobile layer in oscillatory sheet flow Tests were carried out with sand, PVC, and acrylic granules in an oscillatory water tunnel Velocities were measured with a laser Doppler anemometer, and concentrations were measured with conductivity probes The measurements of mobile layer thickness appear to confirm the importance of the parameter S = (ρU o ω/(ρ s -ρ)g) At low values of this parameter the measurements tend toward the unique curve which would be expected if pressure gradient and inertia effects were negligible On the other hand, the velocity profile measurements show some influence of pressure gradient and inertia even for values of S as low as 012 At higher values of S a type of plug flow is observed in which the sediment comes to rest as the flow reverses and then begins to move again as a solid block

A sigma-coordinate primitive equation model for studying the circulation in the South Atlantic. Part I: Model configuration with error estimates

Abstract: This paper describes the configuration of a topography-following (sigma) coordinate, numerical ocean model for studying the circulation in the South Atlantic. An analysis is performed (i) to ensure that the model configuration does not introduce a numerical bias in the model solution and (ii) to give estimates of numerical errors. The model is the Semi-spectral Primitive Equation Model (SPEM) from Rutgers University (Haidvogel et al., 1991). Two important issues relating to the sigma-coordinate are investigated: the pressure gradient calculation and the diffusion of tracers. Errors in the pressure gradient calculation are investigated by simulating an ocean at rest, and the choice is made to reduce errors by smoothing the bathymetry. A smoothing criterion is derived that permits a limitation of the errors in the pressure gradient calculation to an acceptable level (i.e. maximum errors on velocities below a millimeter per second). It is applied to define the model bottom topography. Errors in the tracer fields, induced by a diffusion scheme operating along constant sigma surfaces, generates large unrealistic velocities (of the order of 10 cm/s). A rotation of the diffusion tensor into geopotential coordinates is proposed. Tests show that errors are then reduced to an insignificant level. The rotation of the diffusion tensor is therefore retained. The numerical treatment of the open boundaries and the flux conditions that yields the most realistic circulation is also described. Open boundary conditions are based on radiation conditions and relaxation to climatology. They appear to be numerically robust, and to be able to bring into the South Atlantic basin the necessary information from the outer oceans. A configuration of the SPEM model to study the large scale circulation in the South Atlantic is then obtained. Errors due to model configuration are shown to be small compared to the signal one wants to simulate, and their spatial pattern is known, which will facilitate the interpretation of the model simulations presented in following papers.

H-mode pedestal characteristics, ELMs, and energy confinement in ITER shape discharges on DIII-D

Abstract: The H-mode confinement enhancement factor, H, is found to be strongly correlated with the height of the edge pressure pedestal in ITER shape discharges. In discharges with type I edge-localized modes (ELMs) the pedestal pressure is set by the maximum pressure gradient before the ELM and the width of the H-mode transport barrier. The pressure gradient before type I ELMs is found to scale as would be expected for a stability limit set by ideal ballooning modes, but with values significantly in excess of that predicted by stability code calculations. The width of the H-mode transport barrier is found to scale equally well with pedestal or . The improved H value in high- discharges may be due to a larger edge pressure gradient and wider H-mode transport barrier consistent with their higher edge ballooning mode limit. Deuterium puffing is found to reduce H, which is consistent with the smaller pedestal pressure which results from the reduced barrier width and critical pressure gradient. Type I ELM energy loss is found to be proportional to the change in the pedestal energy.

General wellbore flow model for horizontal, vertical, and slanted well completions

Abstract: A general wellbore flow model, which incorporates not only frictional, accelerational, and gravitational pressure drops, but also the pressure drop caused by inflow, is presented in this paper. The new wellbore model is readily applicable to different wellbore perforation patterns and well completions, and can be easily incorporated in reservoir simulators or analytical reservoir inflow models. Three dimensionless numbers, the accelerational to frictional pressure gradient ratio R{sub af}, the gravitational to frictional pressure gradient ratio R{sub gf}, and the inflow-directional to accelerational pressure gradient ratio R{sub da}, have been introduced to quantitatively describe the relative importance of different pressure gradient components. For fluid flow in a production well, it is expected that there exist three different flow regions along the wellbore, the laminar flow region, the partially-developed turbulent flow region, and the fully-developed turbulent flow region. For wellbore flow with uniform influx, R{sub af} in the laminar flow region is a constant which is only dependent on fluid properties, inflow rate and pipe ID, but independent of axial location and pipe roughness; R{sub af} in the fully-developed turbulent flow region is related to the axial location and pipe geometry (pipe ID and pipe roughness) and may be independent of themore » fluid properties and inflow rate; whereas R{sub af} in the partially- developed turbulent flow region depends on location, pipe geometry, fluid properties and inflow rate. It is found that the influence of either inflow or outflow depends on the flow regime present in the wellbore. It is recommended that the new wellbore flow model be included in wellbore-reservoir coupling models to achieve more accurate predictions of pressure drop and inflow distribution along the wellbore as well as the well production or injection rates.« less

The effects of a favourable pressure gradient and of the Reynolds number on an incompressible axisymmetric turbulent boundary layer. Part 2. The boundary layer with relaminarization

Abstract: This is an experimental investigation of two turbulent boundary layers (cases 2 and 4) where the streamwise negative pressure gradient changes mean properties of the flow, e.g. mean velocity profiles and skin friction, so that they display laminar-like behaviour. The maximum acceleration parameter K ≤ 4 x 10 -6 and the starting value of the Reynolds number is 862 or 2564. Relaminarization occurs in both boundary layers as a gradual change of the turbulence properties and is not catastrophic. Retransition, however, is a fast process due to the remaining turbulence structure and may be compared with bypass transition. Together with an extensive investigation of the turbulence structure as in the companion paper, Part 1, which describes two cases (1 and 3) of boundary layers which remain turbulent, spectra and integral length scales for all four boundary layers are discussed

Numerical investigation of the turbulent boundary layer over a bump

Abstract: Large-eddy simulation (LES) has been used to calculate the flow of a statistically two-dimensional turbulent boundary layer over a bump. Subgrid-scale stresses in the filtered Navier–Stokes equations were closed using the dynamic eddy viscosity model. LES predictions for a range of grid resolutions were compared to the experimental measurements of Webster et al. (1996). Predictions of the mean flow and turbulence intensities are in good agreement with measurements. The resolved turbulent shear stress is in reasonable agreement with data, though the peak is over-predicted near the trailing edge of the bump. Analysis of the flow confirms the existence of internal layers over the bump surface upstream of the summit and along the downstream trailing at plate, and demonstrates that the quasi-step increases in skin friction due to perturbations in pressure gradient and surface curvature selectively enhance near-wall shear production of turbulent stresses and are responsible for the formation of the internal layers. Though the flow experiences a strong adverse pressure gradient along the rear surface, the boundary layer is unique in that intermittent detachment occurring near the wall is not followed by mean-flow separation. Certain turbulence characteristics in this region are similar to those previously reported in instantaneously separating boundary layers. The present investigation also explains the driving mechanism for the surprisingly rapid return to equilibrium over the trailing flat plate found in the measurements of Webster et al. (1996), i.e. the simultaneous uninterrupted development of an inner energy-equilibrium region and the monotonic decay of elevated turbulence shear production away from the wall. LES results were also used to examine response of the dynamic eddy viscosity model. While subgrid-scale dissipation decreases/increases as the turbulence is attenuated/enhanced, the ratio of the (averaged) forward to reverse energy transfers predicted by the model is roughly constant over a significant part of the layer. Model predictions of backscatter, calculated as the percentage of points where the model coefficient is negative, show a rapid recovery downstream similar to the mean-flow and turbulence quantities.

Net flow of compressible viscous liquids induced by travelling waves in porous media

Abstract: The influence of ultrasonic radiation on the flow of a liquid through a porous medium is analyzed. The analysis is based on a mechanism proposed by Ganiev et al. according to which ultrasonic radiation deforms the walls of the pores in the shape of travelling transversal waves. Like in peristaltic pumping, the travelling transversal wave induces a net flow of the liquid inside the pore. In this article, the wave amplitude is related to the power output of an acoustic source, while the wave speed is expressed in terms of the shear modulus of the porous medium. The viscosity as well as the compressibility of the liquid are taken into account. The Navier–Stokes equations for an axisymmetric cylindrical pore are solved by means of a perturbation analysis, in which the ratio of the wave amplitude to the radius of the pore is the small parameter. In the second-order approximation a net flow induced by the travelling wave is found. For various values of the compressibility of the liquid, the Reynolds number and the frequency of the wave, the net flow rate is calculated. The calculations disclose that the compressibility of the liquid has a strong influence on the net flow induced. Furthermore, by a comparison with the flow induced by the pressure gradient in an oil reservoir, the net flow induced by a travelling wave can not be neglected, although it is a second-order effect.

A General Pressure Gradient Formulation for Ocean Models. Part I: Scheme Design and Diagnostic Analysis

Abstract: A Jacobian formulation of the pressure gradient force for use in models with topography-following coordinates is proposed. It can be used in conjunction with any vertical coordinate system and is easily implemented. Vertical variations in the pressure gradient are expressed in terms of a vertical integral of the Jacobian of density and depth with respect to the vertical computational coordinate. Finite difference approximations are made on the density field, consistent with piecewise linear and continuous fields, and accurate pressure gradients are obtained by vertically integrating the discrete Jacobian from sea surface. Two discrete schemes are derived and examined in detail: the first using standard centered differencing in the generalized vertical coordinate and the second using a vertical weighting such that the finite differences are centered with respect to the Cartesian z coordinate. Both schemes achieve second-order accuracy for any vertical coordinate system and are significantly more a...

The characterization of multiphase fluid transport in a porous solid by pulsed gradient stimulated echo nuclear magnetic resonance

Abstract: Pulsed magnetic field gradient stimulated echo nuclear magnetic resonance (NMR) measurements are reported for the steady-state flow and diffusion of two and three phases (water, dodecane, N2 gas) within a sample of a Fontainebleau sandstone. The stimulated echo dependence on the gradient pulse area, q, is used to derive the displacement probability, PΔ(X) for fixed observation times Δ, with the displacements X being measured along the macroscopic pressure gradient. An extensive range of NMR experiments was carried out, starting with single-phase flow of either water (an aqueous solution of NaCl 3% w/v) or oil (dodecane) for various relative saturation states. Following these experiments, PΔ(X) were acquired for water and oil when both phases were forced to flow through the sandstone. Finally, NMR measurements were performed in which three phases (oil, water and N2 gas) were flowing simultaneously. Using the NMR data it was possible to evaluate the physical importance of parameters such as wettability, spr...

Lagrangian accelerations in geostrophic turbulence

Abstract: A distinctive property of Lagrangian accelerations in geostrophic turbulence is that they are governed by the large and intermediate scales of the flow, both in time and space, so that the inertial part of the dynamics plays a much larger role than in three-dimensional turbulence where viscous effects are stronger. For the case of geostrophic turbulence on a β-plane, three terms contribute to the Lagrangian accelerations: the ageostrophic pressure gradient which often is the largest term, a meridional acceleration due to the β-effect, and an acceleration due to horizontally divergent ageostrophic motions. Both their spectral characteristics and patterns in physical space are studied in this paper. In particular the total accelerations field has an inertial spectrum slope which is identical to the geostrophic velocity field inertial slope. The accelerations gradient tensor is shown to govern the topology of quasi-geostrophic stirring and transport properties. Its positive eigenvalues locate accurately the position of extrema of potential vorticity gradients. The three-dimensional distribution of tracer gradients is such that the vertical distribution is entirely constrained by the horizontal one, while the reverse is not true. We make explicit analytically their dependence on the three-dimensional accelerations gradient.

A General Pressure Gradient Formulation for Ocean Models. Part II: Energy, Momentum, and Bottom Torque Consistency

Abstract: A new formulation of the pressure gradient force for use in models with topography-following coordinates is proposed and diagnostically analyzed in Part I. Here, it is shown that important properties of the continuous equations are retained by the resulting numerical schemes, and their performance in prognostic simulations is examined. Numerical consistency is investigated with respect to global energy conservation, depth-integrated momentum changes, and the representation of the bottom pressure torque. The performances of the numerical schemes are tested in prognostic integrations of an ocean model to demonstrate numerical accuracy and long-term integral stability. Two typical geometries, an isolated tall seamount and an unforced basin with sloping boundaries, are considered for the special case of no external forcing and horizontal isopycnals to test numerical accuracy. These test problems confirm that the proposed schemes yield accurate approximations to the pressure gradient force. Integral c...

Large-Eddy Simulation of the Diurnal Cycle of Deep Equatorial Turbulence

Abstract: The deep diurnal cycle of turbulence at the equator is studied using the technique of large-eddy simulation (LES). Based on a scale-separation hypothesis, the LES model includes the following large-scale flow terms: the equatorial undercurrent (EUC), zonal pressure gradient, upwelling, horizontal divergence, zonal temperature gradient, and mesoscale eddy forcing terms for the zonal momentum and the heat equations. The importance of these terms in obtaining a quasi-equilibrium boundary layer solution is discussed. The model is forced with a constant easterly wind stress and diurnal cooling and heating. It is found that boundary-layer turbulence penetrates as deep as 50 m below the mixed layer during nighttime cooling. The diurnal variation of turbulence dissipation and mixed layer depth are within the range of observations. The gradient Richardson number (Ri) of the mean flow shows a diurnal cycle but the amplitudes decrease with depth. Within the mixed layer and just below the layer, Ri can be lo...

Azimuthal pressure gradient as driving force of substorm currents

Abstract: We have studied the azimuthal pressure gradient in the central plasma sheet during substorms using plasma and magnetic field data obtained by the AMPTE/IRM satellite at nightside in radial distances of 9–15 RE. The pressure gradient is statistically estimated for the interval when the magnetic field shows a dipolar configuration (elevation angle >45°). It is found that by this criterion, most data are obtained during and after the passage of high-speed ion flow in the vicinity of the neutral sheet during magnetically active times. We show that there is an azimuthal gradient of plasma pressure in the dipolar field region. The pressure gradient can drive a substantial amount of field-aligned current (4.1×105 A per 2-hour local time sector). We suggest that this current is a source of the substorm current system after the high-speed ion flow stops.

Unsteady blood flow in a helically symmetric pipe

Abstract: Fully developed flow in a helical pipe is investigated with a view to modelling blood flow around the commonly non-planar bends in the arterial system. Medical research suggests that the formation of atherosclerotic lesions is strongly correlated with regions of low wall shear and it has been suggested that the observed non-planar geometry may result in a more uniform shear distribution. Helical flows driven by an oscillating pressure gradient are studied analytically and numerically. In the high-frequency limit an expression is derived for the second-order steady flow driven by streaming from the Stokes layers. Finite difference methods are used to calculate flows driven by sinusoidal or physiological pressure gradients in various geometries. Possible advantages of the observed helical rather than planar arterial bends are discussed in terms of wall shear distribution and the inhibition of boundary layer separation.

Baroclinic circulation generation on shock accelerated slow/fast gas interfaces

Abstract: Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimensions, for a planar “slow–fast” interface, we present analytical results for σ, the circulation deposited per unit unshocked interface length, within the regular refraction regime. The parameters that describe the interaction are the Mach number (M), the density ratio of the two gases (η, η<1), the local angle between the shock and the interface (α), and the ratio of specific heats of the two gases (γ0,γb). For weak shocks σ scales as σ∝(1−η−1/2)ξ(M)sin α and for strong shocks σ→K(η,α,γ)/(1−ξ(M)). For scaling purposes, the gases are assumed to have the same γ. K(η,α,γ) is a function of the density ratio, the interface angle, and the ratio of specific heats γ [Eq. (4.6)] and ξ(M) is the normalized pressure gradient across the shock. The planar interface approach is used to find formulas to calculate the total circulation deposited on sinusoidal interfaces. To validate the formulas, numerical simulations of the compressible Euler equations were made using a second-order Godunov code. Simulations were done for 1.05⩽M⩽3.0 and η=0.14, 0.33 and 0.65.

Ballooning mode stability for self-consistent pressure and current profiles at the H-mode edge

Abstract: The edge pressure gradient (H-mode pedestal) for computed equilibria in which the current density profile is consistent with the bootstrap current may not be limited by the first-regime ballooning limit. The transition to second stability is easier for higher elongation, intermediate triangularity, larger aspect ratio, pedestal at larger radius, narrower pedestal width, higher , and lower collisionality.